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M1N  CIRCULATING 

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UNIVERSITY  OF  ILLINOIS 

Agricultural  Experiment  Station 


BULLETIN  No.  272 


RATE  OF  MILK  SECRETION  AS  AFFECTED 

BY  ADVANCE  IN  LACTATION 

AND  GESTATION 

Correction  of  Yield,  Within  a  Lactation  Period, 
for  Length  of  Record  and  for  Pregnancy 

BY  W.  L.  GAINES  AND  F.  A.  DAVIDSON 


URBANA,  ILLINOIS,  JANUARY,  1926 


CONTENTS 

PAGE 

INTRODUCTION. 3 

SOURCE  OF  DATA  AND  STATISTICAL  TREATMENT 5 

EQUATION  OF  THE  LACTATION  CURVE 7 

RESULTS ... 10 

Farrow  Cows 10 

Gestating  Cows 13 

Calf  Carried  Five  Months  or  Less 13 

Calf  Carried  More  Than  Five  Months 14 

Age  and  Productive  Level 18 

Fat  Percentage 26 

DISCUSSION 27 

CORRECTION  FACTORS  FOR  LENGTH  OF  RECORD 33 

CORRECTION  FACTORS  FOR  PREGNANCY 33 

SUMMARY 34 

LITERATURE  CITED 36 

Note. — An  extract  from  the  material  presented  in  this  bulletin  appeared  in  the 
January,  1926,  number  of  the  Journal  of  General  Physiology:  Gaines,  W.  L.,  and 
Davidson,  F.  A.  The  effect  of  advance  in  lactation  and  gestation  on  mammary 
activity.  Jour.  Gen.  Physiol.  9,  325-332.  1926. 


RATE  OF  MILK  SECRETION  AS  AFFECTED 

BY  ADVANCE  IN  LACTATION 

AND  GESTATION 

Correction  of  Yield,  Within  a  Lactation  Period, 
for  Length  of  Record  and  for  Pregnancy 

By  W.  L.  GAINES,  Chief  in  Milk  Production,  and  F.  A.  DAVIDSON, 
First  Assistant  in  Dairy  Husbandry 

INTRODUCTION 

The  practice  of 'breeders  of  dairy  cattle  in  officially  testing  their 
cows  for  production  differs  in  several  ways.  Two  such  differences  that 
appear  in  the  published  data  of  the  Advanced  Registry  are  the  length 
of  record  and  pregnancy.  The  length  of  record — that  is,  the  number  of 
days  covered  by  the  record — is  of  course  a  major  factor  in  the  amount 
of  milk  produced.  Pregnancy,  or  the  length  of  time  a  calf  is  carried  by 
the  cow  during  the  production  period,  is  also  a  factor  affecting  milk 
yield.  In  order  properly  to  compare  records  in  which  these  factors 
differ,  it  is  necessary  to  know  something  of  the  quantitative  relation 
between  the  length  of  the  record  and  the  amount  of  milk  produced, 
and  of  the  quantitative  effect  of  pregnancy  thruout  its  course  upon 
milk  yield. 

The  total  amount  of  milk  yielded  during  any  lactation  period — 
that  is,  from  parturition  to  the  end  of  lactation  or  until  recurrence  of 
parturition — is  greatly  influenced  by  the  length  of  time  over  which  lac- 
tation extends.  The  length  of  lactation,  in  turn,  is  profoundly  influenced 
by  the  service  period  (parturition  to  conception) .  Ellinger1  found  in  the 
Red  Danish  breed  a  high  correlation  between  length  of  lactation  and 
service  period:  r  =  .943  zh  .005.  This  coefficient  affords  a  numerical 
measure  of  the  closeness  of  the  relationship  between  gestation  and  the 
termination  of  lactation.  A  similar  coefficient  of  correlation  between 
total  yield  and  length  of  lactation  has  not  been  derived,  so  far  as  we  are 
aware,  but  Hammond  and  Sanders2  found  the  correlation  between 
service  period  and  yield  to  be  r  =  .33  ±  .016.  They  derived  the  equa- 
tion y  =  8500  —  4250e~'c  **  as  expressing  the  relation  between  ser- 
vice period  and  yield  for  the  lactation  (y  =  milk  yield  in  pounds  and 
x  =  service  period  in  days).  According  to  this  equation  the  lactation 
yield  would  be  4,250  pounds  if  conception  occurred  at  once,  and 
would  reach  8,500  pounds  as  a  limit  if  breeding  were  indefinitely  post- 


4  BULLETIN  No.  272  [January, 

poned.  From  this  equation  Hammond  and  Sanders  have  derived  a  set 
of  correction  factors  using  a  100-day  service  period  as  a  base.  It  will  be 
apparent  that  their  correction  is  intended  to  measure  the  combined 
effect  of  length  of  record  and  pregnancy  on  the  lactation  yield.  Their 
results  are  based  on  1,410  records  of  Shorthorn  cows,  chiefly  nonpedigree, 
obtained  from  a  milk  recording  society  in  England. 

The  great  majority  of  the  published  records  of  cows  in  the  United 
States  are  for  a  partial  lactation  period  starting  shortly  after  calving 
and  continuing  for  tunes  varying  from  7  to  365  days.  Effective  breeding 
may  occur  a  few  days  after  parturition  or  may  be  so  delayed  that  the 
calf  is  carried  any  time  from  280  days  (full  term)  to  0  days  during  the 
record  period.  The  length  of  the  record  and  of  pregnancy  are  two  of 
the  many  variable  factors  which  affect  the  magnitude  of  the  published 
record  of  production.  We  expect  a  record  for  200  days  to  be  greater 
than  that  for  100  days,  but  not  twice  as  great.  In  records  of  the  same 
length  we  expect  the  production  of  the  farrow  cow  to  exceed  that  of  the 
gestating  cow  to  some  extent.  A  more  definite  quantitative  expression 
of  these  relations,  if  they  are  not  too  irregular,  may  permit  a  scheme  of 
correction  that  will  reduce  the  record  to  its  equivalent  under  standard 
conditions. 

The  relation  between  the  7-day  and  365-day  records  of  the  Holstein 
breed  has  been  studied  by  Yapp3  and  by  Gowen  and  Gowen.4  Yapp 
considered  all  ages  together  and  found  the  correlation  between  the  milk 
records  for  the  two  periods  within  the  same  lactation  to  be  r  =  .702  ±  .01. 
Gowen  and  Gowen  find  this  coefficient  reduced  to  about  .6  when  confined 
to  narrow  age  limits.  The  latter  authors  have  derived  a  complete  series 
of  prediction  equations  or  correction  formulae  to  express,  for  various 
ages,  the  quantitative  relations  between  the  two  records.  These  formulae 
are  based  on  a  linear  equation,  y  —  a  +  bx,  where  y  is  the  yield  to  be 
expected  from  the  record,  x,  which  is  known. 

We  are  concerned  in  this  study  more  particularly  with  records 
of  varying  length  in  the  same  lactation,  say  from  200  to  365  days,  during 
which  the  calf  is  carried  varying  lengths  of  tune.  Of  particular  interest 
is  the  relation  between  the  305-day  and  365-day  records,  since  these 
periods  constitute  a  common  basis  of  distinction  in  the  official  testing 
of  several  of  our  dairy  breeds. 

It  is  obvious  that  milk  yield  is  the  result  of  the  rate  of  milk  secre- 
tion. It  is  convenient  to  use  the  term  lactation  curve  for  either  the 
curve  representing  the  true  theoretical  rate  of  milk  secretion  or  that  rep- 
resenting the  approximate  rate  of  secretion  shown  by  the  monthly 
yields;  the  context  and  notation  will  indicate  the  usage.  If  an  equation 
for  the  lactation  curve  can  be  determined,  then  the  equation  may  be 
made  the  basis  for  estimating  the  yield  for  any  portion  of  the  period 


1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION  5 

covered.  That  the  lactation  curve  is  capable  of  simple  mathematical 
expression  adapted  to  the  purpose  at  hand  is  indicated  by  various  in- 
vestigations. Sturtevant,5  in  an  early  study  of  the  decrease  in  milk 
yield  with  advance  in  lactation,  reached  the  conclusion  that  ".  .  .  . 
the  natural  falling  off  in  milk  for  each  month  from  calving,  is  about 
nine  percent  of  the  yield  of  the  preceding  month."  His  month  is  a 
thirty-day  period  and  his  conclusion  is  based  on  the  records  of  83  cows 
(45  Ayrshires,  3  Jerseys,  35  natives),  for  210  lactations  during  the  years 
1866-1880,  in  a  private  herd.  Later  investigators  have,  in  general, 
confirmed  the  applicability  of  this  method  of  expressing  the  lactation 
curve. 

The  above  relation  may  raise  the  presumption  that  the  rate  of  milk 
secretion  is  continually  decreasing  at  a  rate  proportional  to  its  value 
at  the  moment,  since  changes  of  such  nature  are  of  common  occurrence 
in  natural  phenomena  and  lead  to  precisely  such  a  relation  as  found 
by  Sturtevant.  Brody,  Ragsdale,  and  Turner6  have  recently  shown 
that  the  decline  in  milk  yield  which  occurs  with  advance  in  lactation  con- 
forms to  the  law  of  certain  chemical  reactions  and  may  be  expressed 
as  "M t  =  M0e~ht  where  Mt  =  milk  production  during  any  month,  t; 
M0  is  the  theoretical  value  of  the  milk  flow  at  the  time  of  parturition; 
e  and  k  have  the  usual  meaning."  Their  equation  is  another  way  of 
expressing,  in  a  more  general  and  precise  mathematical  form,  the  con- 
clusion of  Sturtevant.  Regardless  of  the  inferences  drawn  by  these 
authors  as  to  the  chemical  processes  of  milk  secretion,  it  is  evident 
from  the  data  which  they  present  that  the  equation  gives  a  curve  con- 
forming well  with  the  observed  monthly  yields,  at  least  within  certain 
time^  limits.* 

SOURCE  OF  DATA  AND  STATISTICAL  TREATMENT 

The  data  of  the  present  paper  are  taken  from  the  published  records 
of  the  American  Guernsey  Cattle  Club:  namely,  Vols.  33  and  34,  and 
No.  1  of  Vol.  35,  of  the  Advanced  Register.  The  published  record  in- 
cludes for  each  cow  the  date  of  birth,  date  of  calving,  date  of  effective 
service,  and  the  milk  and  fat  yields  by  calendar  months.  For  the  pur- 
pose of  classification  the  printed  pages  of  the  record  were  removed  and 
backed  with  sheets  of  gummed  paper  board.  The  sheets  were  then  so 
cut  as  to  give  the  record  of  each  cow  on  a  card  of  convenient  size.  The 


°A  significant  deviation  of  the  observed  values  from  the  theoretical  during  the 
first  month  or  two  of  lactation  is  noted  by  the  authors,  and  further  treated  on  a  chemi- 
cal basis  in  a  later  paper.7  For  simplicity  and  for  the  purpose  of  the  present  dis- 
cussion, this  discrepancy  may  be  ignored  for  the  time  being,  to  be  mentioned  again 
later  (pages  31  and  32). 


6  BULLETIN  No.  272  [January, 

work  of  classification  was  thus  facilitated  and  possible  errors  of  tran- 
script were  avoided. 

For  the  record  of  each  cow  additional  data  were  computed  and  re- 
corded on  the  card,  as  follows:  age  of  cow  at  calving  (where  necessary); 
time  in  days  from  calving  to  the  beginning  of  the  first  full  calendar- 
month  record  of  milk  and  fat  yield;  and  the  time  in  days  from  calving 
to  conception.  All  records  in  which  the  time  from  calving  to  the  be- 
ginning of  the  first  full  calendar-month  record  exceeded  60  days,  and 
all  records  in  which  essential  data  were  lacking  were  discarded.  A 
total  of  4,522  records  were  used,  including  entries  and  re-entries. 

The  records  were  then  grouped  with  respect  to  the  time  of  con- 
ception in  days  after  calving,  intervals  of  30.5  days,  1-31  days,  32-61 
days,  62-92  days,  and  so  on,  to  a  final  group  of  farrow  cows,  that  is, 
cows  not  bred  at  366  days  after  calving,  being  used.  Conception  is  re- 
garded as  occuring  at  the  middle  of  the  intervals,  that  is,  at  .5  month 
after  calving  in  the  first  group;  at  1.5  months  in  the  second;  and  so  on. 
This  classification  was  made  in  order  to  study  the  effect  of  pregnancy. 

In  order  to  study  the  effect  of  advance  in  lactation,  each  of  the 
above  thirteen  groups  was  further  separated  into  two  subgroups:  the 
first  subgroup  (a)  including  those  records  in  which  1  to  30  days  elapsed 
between  calving  and  the  beginning  of  the  first  full  calendar-month 
record;  and  the  second  subgroup  (b)  including  those  in  which  31*  to  60 
days  so  elapsed.  The  average  yield  for  subgroup  (a)  for  the  first  full 
calendar-month  is  taken  to  represent  the  yield  for  a  month,  the  mid- 
point of  which  is  one  month  after  calving.  This  gives  the  first  observa- 
tion for  the  lactation  curve.  The  second  observation  for  the  lactation 
curve  (the  yield  for  a  month,  the  mid-point  of  which  is  two  months 
after  calving)  is  the  average  yield  for  the  second  calendar  month  of 
subgroup  (a)  combined  with  the  first  calendar  month  of  subgroup  (b) ; 
and  so  on  to  the  eleventh  observation  for  the  lactation  curve,  which  is 
the  average  of  the  eleventh  calendar  month  of  subgroup  (a)  combined 
with  the  tenth  calendar  month  of  subgroup  (b).  A  possible  twelfth 
observation  represented  by  the  eleventh  calendar  month  of  subgroup 
(b)  was  not  used. 

The  average  yields  have  been  computed  by  tabulating  the  data  by 
months  in  the  form  of  a  correlation  table,  using  class  intervals  of  5  pounds 
for  fat  and  100  pounds  for  milk.  The  data  for  milk  and  fat  have  been 
converted  to  a  single  expression  representing  the  physiological  equiva- 
lent of  4-percent  milk  on  the  basis  of  gross  energy  value  and  designat- 
ed fat-corrected  milk,  F.C.M.,  in  accordance  with  the  writers'  ideas  pre- 
viously presented.8  The  equation  is  F.C.M.  =  AM  +  15F,  where  M  is 
milk  and  F  is  fat,  and  all  in  the  same  unit  of  weight,  that  is,  the  pound  in 
the  present  data.  The  fat-corrected  milk  is  to  be  regarded  as  an  esti- 


1926}       MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION  7 

mate  of  the  energy  yield  in  terms  of  natural  4-percent  milk  (one  pound 
F.C.M.  =  one  pound  of  4-percent  milk  =  330.62  large  calories). 

EQUATION  OF  THE  LACTATION  CURVE 
If  the  lactation  curve  can  be  expressed  as  a  rate  or  velocity  in  the 
form  of  a  differential  equation,  capable  of  integration,  we  have  then  the 
means  of  computing  the  area  under  the  curve,  or  the  yield  between  any 
time  limits  desired.  As  suggested  in  the  introduction,  it  is  purposed  to 
use  the  equation: 


=  ae~kt  (1) 

dt  W 

in  which  y  =  yield  in  pounds;  t  =  time  in  months  (30.5  days)  from 

dij 
calving;  —  -  represents  the  rate  of  yield  in  pounds  per  month;  a  is  a  con- 

CH 

stant  representing  the  theoretical  initial  rate  of  yield  in  pounds  per 
month;  A;  is  a  constant  representing  the  rate  of  change  per  month  (as 

dii 
proportional  to  -^)  in  the  rate  of  yield  per  month,  the  minus  sign  indi- 

CLL 

eating  that  the  change  is  toward  smaller  and  smaller   values;  and 

e  =  2.71828.a 


aThe  significance  of  the  factor  e  in  the  present  connection  may  be  illustrated  by 
the  familiar  case  of  money  at  simple  and  compound  interest.  Let  a  represent  the 
original  principal  bearing  interest  at  the  rate  of  r  percent  per  annum,  and  let  k  = 
r/100,  that  is,  the  rate  of  interest  expressed  as  a  decimal.  At  the  end  of  t  years  the 
amount  at  simple  interest  is  a  X  (1  +  kt)  that  is,  a+  akt;  but  if  the  interest  is  added 
to  the  principal  every  instant  the  amount  at  the  end  of  I  years  is  a  times  e  to  the  power 
kt,  that  is,  aekt.  In  the  time  required  for  the  principal  to  double  at  simple  interest 
(kt  =  1),  it  would  increase  2.71828  fold  at  true  compound  interest.  If  the  principal 
were  decreasing  in  a  similar  manner,  instead  of  becoming  a  times  ekt,  it  would  become 

a  divided  by  ekt  that  is,   —^,  which  may  be  written  ae~kt. 

Under  the  assumption  that  the  rate  of  yield  is  continuously  decreasing  at  a  rate 
proportional  to  its  value  at  the  moment,  equation  (1)  may  be  derived  thus:  Let 

y'  =  -~-;t\ien— ^-  =  —ky'  where  A;  is  the  constant  of  proportionality.  Multiplying 
by  —f-  we  have  — ^-  =  —  kdt,    and   integrating,  log  y'  =  —  kt  +  C.    If  a  represents 

y  y 

v 
the  initial  rate,  that  is,  a  =  y'  when  t  =  0,  then  C  =  log  a  and  log   -—  =  —  kt. 

•  dij 

Passing  from  logarithms  to  exponentials,  y'  ——,r  —  ae~kt. 

Oil 

The  rate  of  milk  secretion  does  not  change  from  moment  to  moment  between 
milkings  at  the  same  rate  as  required  by  equation  (1)  when  applied  to  longer  periods, 
but  for  the  purpose  in  view,  and  the  gross  periods  which  we  shall  have  to  consider 
in  the  integrated  expression,  we  may  ignore  the  periodic  fluctuations  correlated  with 
the  occurrence  of  milking. 


8  BULLETIN  No.  272  [January, 

Integration  of  (1)  gives: 

fdy  =  f ae~ktdt 
and, 

yfS-«.r*<  +  C  (2) 

Since  we  are  dealing  only  with  yields  following  calving,  y  =  0 
when  t  =  0,  and  we  may  evaluate  C  by  substituting  y  =  0  and  t  =  0 
in  (2),  giving: 


Substituting  this  value  of  C  in  (2), 

y  =  |  (1  -  e-*<)  (3) 

If  we  have  the  constants  a  and  k  of  (1),  then  we  can  compute  y 
(=  yield)  up  to  any  time  or  between  any  time  limits  from  (3). 

From  the  eleven  observations  for  the  lactation  curve  we  have  to 
determine  values  for  a  and  k  of  (1).  Clearly,  the  observations  are  to  be 
taken  as  definite  integrals  of  (1);  the  first  observation  corresponds  to 

/•l.S  /»2.5 

the  theoretical  value  I     ae~ktdt;  the  second,  to  I     ae~htdt;  and  so  on, 

Jo.S  J  1.6 

/11.5 
ae~ktdt. 
0.5 

The  ratio  between  the  theoretical  yield  for  any  month  and  that 
of  the  next  preceding  month  which  will  satisfy  equation  (1)  is  a  constant, 
e~k.  Let  ym  designate  the  yield  for  a  month,  and  let  time  (i)  be  counted 
to  the  middle  of  the  month  concerned.3  Consider  any  two  consecutive 
months,  ymi  (where  t  =  ri)  and  yma  (where  t  =  n  +  1)  and  n  is  any 
assigned  value  (from  .5  up),  then: 

ymi    =  |  ["(?-*<»  -  -5)    _   g-Hn  +  .5)  j  (4) 

and, 


The  ratio  of  the  yield  for  any  month  to  that  of  the  next  preceding 
month  is  then,  from  (5)  and  (4)  : 


"In  the  equations  of  the  lactation  curves,  which  henceforth  are  expressed  in 
terms  of  ym  and  t,  it  will  be  understood  that  t  is  reckoned  to  the  middle  of  the  month. 


1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION  9 

£  [e-Hn+.S)   _   e-i(n+1.5)    1 

£_.  _  U  -  J  _  ^  (6)b 

"*»  ^[~e-*(n-  .5)    _   g-«(n+  .5) 

Equation  (4)  may  be  written  : 

4.         —  _  |/>.5Jt  _   />—  .5fc  |  />—  in 

ym'  "  k  le          r 

and  returning  to  the  general  form, 

ym  =  Ae~kt  (7) 

where, 


and  transposing, 

«  =  ^    e.6fe  _  e-.5*  (8) 

Equation  (7)  is  readily  applied  to  the  observed  values.  Ae~k 
corresponds  to  the  first  observation;  Ae~2k,  to  the  second;  Ae~3k,  to 
the  third  and  so  on.  We  may  write  in  place  of  (7), 

logio  ym  =  logio  A  —  kt  logio  e 

and  in  this  form  the  equation  is  linear  in  logio  ym  and  t;  also  in 
logio  A  and  k.  From  a  straight  line  fitted  graphically  or  by  the  method 
of  least  squares  to  logio  of  the  observed  yields,  values  for  A  and  k  of  (7) 
may  be  determined  (see  Running,9  Formula  V).  The  constant,  k,  of  (1) 
is  the  same  as  that  of  (7),  and  a  of  (1)  is  derived  from  A  of  (7)  thru 
equation  (8).  When  k  is  small,  as  in  the  present  data,  A  and  a  are 
practically  equal;  thus,  when  k  =  .05,  a  =  .999898A.  While  equations 
(1)  and  (7)  are  by  no  means  mathematically  identical  they  may,  for 
practical  purposes,  be  used  interchangeably  to  express  the  lactation 
curve  (cf.  page  4)  outside  the  influence  of  pregnancy.  A  similar  usage 
of  equations  (9)  and  (10)  in  describing  the  lactation  curve  where  the 
influence  of  pregnancy  must  be  recognized,  is  also  roughly  justified,  as 
will  appear  below. 

In  the  case  of  gestating  cows,  equation  (7)  is  not  sufficient  to 
describe  the  lactation  curve  after  pregnancy  becomes  somewhat  ad- 


bThe  constant  percentage  decrease  from  month  to  month,  previously  referred 
to  in  Sturtevant's  work,  corresponds  roughly  to  the  constant  of  proportionality,  k, 
in  ratio  (6),  and  100  k  is  approximately  equal  to  the  percentage  decline.  The  approxi- 
mation holds  only  for  small  values  of  k.  When  k  =  .045,  e~k  =  .9560  and  the  per- 
centage decline  is  4.40;  whenfc  =  .090,  e~k  =  .9139  and  the  percentage  decline  is  8.61. 


10  BULLETIN  No.  272  [January, 

vanced.  It  is  necessary  to  add  a  term  to  describe  the  decrease  in  yield 
associated  with  pregnancy.  The  equation  used  for  cows  in  advanced 
pregnancy  in  its  differential  form  is: 

^  =  ae~kt  -  &e*«-<> 
dt 

and  as  applied  to  the  observed  values, 

ym  =  Ae~kt  -  Be  *e«-«>  (10) 

The  first  terms  on  the  right  in  (9)  and  (10)  are  the  same  as  used  in 
(1)  and  (7)  respectively.  In  the  second  terms,  c  is  time  in  months  from 
calving  to  conception  and  is  determined,  for  the  groups  where  used,  as 
indicated  under  the  description  of  the  statistical  classifications  (page  6). 
Consequently,  t  —  c  is  time  in  months  from  conception. 

Equation  (10)  has  been  applied  to  the  data  of  gestating  cows  in 
those  groups  where  conception  occured  5.5  months  after  calving  or 
earlier.  The  constants  A  and  k  of  the  first  term  have  been  determined 
from  the  observations  up  to  and  including  t  —  c  =  5.5  by  the  method 
above  described.  The  constants  B  and  K  of  the  second  term  have  been 
derived  by  a  similar  method  from  the  deviations  of  the  observed  values 
from  the  calculated  values  of  Ae~kt,  beyond  t  —  c  =  5.5.  It  may  be 
assumed  that  the  deviations  are  due  to  pregnancy,  and  it  will  be  apparent 
from  the  value  given  c  that  t  —  c,  as  used  in  (10),  measures  time  from 
conception  to  the  middle  of  the  month  of  pregnancy.  Under  these  con- 
ditions, it  may  be  shown  that: 


(11) 


From  the  values  found  for  the  second  term  of  (10),  the  second 
term  of  (9)  may  be  derived  by  application  of  (11).  It  is  evident  we  may 
substitute  the  time  in  months  during  which  the  calf  is  carried  for  t  —  c 
of  (9),  and  by  integration  of  the  second  term  alone  compute  the  effect 
of  pregnancy  during  any  portion  of  the  gestation  period. 

RESULTS 

Farrow  Cows.-  —  In  presenting  the  results  for  farrow  cows,  we  may 
confine  our  attention  to  the  observed  and  calculated  data  from  (7), 
ym  =  Ae~kt,  bearing  in  mind  that  ym  =  yield  for  a  month,  and  t  =  time 
in  months  from  calving  reckoned  to  the  middle  of  the  month  under 
consideration.  The  numerical  data  for  the  farrow-cow  group  are  given 


1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION 


11 


in  Table  1.  Equation  (7)  has  been  fitted  by  the  method  of  least  squares 
to  the  three  sets  of  observed  data — milk  yield,  fat  yield,  and  fat-cor- 
rected milk  yield.  The  purpose  in  presenting  the  three  sets  of  data  is  to 
show  the  interrelation  between  them  and  the  degree  to  which  the  cal- 
culated values  conform  in  each  case  to  the  observed  values.  Since  the 
curves  have  been  fitted  by  the  method  of  least  squares,  the  root-mean- 
square  errors  give  a  comparative  measure  of  the  agreement  between 
observed  and  calculated  values  for  the  three  sets  of  data,  if  allowance 
is  made  for  the  relative  magnitudes  involved.  Such  an  allowance  can 
be  fairly  made  by  weighting  the  error  for  each  curve  by  the  reciprocal 
of  its  A.  This  has  been  done  in  the  last  line  of  Table  1.  Considering 
the  weighted  error  of  the  F.C.M.  curve  as  100,  the  error  for  the  fat  curve 
is  accordingly  138  and  that  for  milk,  147. 

The  observed  data  and  fitted  curves  of  Table  1  are  shown  graph- 
ically in  Fig.  1,  the  data  for  fat  yield  being  multiplied  by  25  to  bring 


Fig  1-  Date  Of  MM  Sacretk 
otf  cowj) 


Tim*  In  Months  from  Calring  ft) 


them  into  approximation  with  the  other  data.  A  very  good  agreement 
is  evident  for  all  three,  but  in  so  far  as  equation  (7)  may  represent  the 
underlying  law  governing  the  change  in  rate  of  milk  secretion  with 
advance  in  lactation,  it  seems  that  the  energy  yield  (F.C.M.)  is  more 
amenable  than  either  milk  yield  or  fat  yield. 

It  may  be  noted  from  Fig.  1  that  the  curve  for  milk  yield  declines 
most  rapidly,  that  for  fat  yield  least  rapidly,  while  that  for  F.C.M.  is 
intermediate.  There  is  quite  a  marked  progressive  change  in  the 
composition  of  the  milk  of  Guernsey  cows  with  advance  in  lactation,  as 
shown  by  the  fat  percentage  data  in  Table  1.  The  F.C.M.  values  take 
account  of  the  change  in  concentration  of  the  fat  and  the  accompanying 
solids-not-fat  in  the  milk.  From  the  equations  in  Table  1  it  will  be  seen 
that  the  rapidity  of  decline  of  the  curves  in  Fig.  1  varies  with  A;;  the 
larger  k,  the  more  rapid  the  decline. 


12 


BULLETIN  No.  272 


[January, 


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1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION  13 

The  figures  for  milk  yield  in  Table  1  confirm  the  results  in  the  work 
of  Brody  et  al6  previously  mentioned,  as  well  as  those  presented  in  a 
later  paper10  in  which  the  investigators  show  a  value  for  A  (as  used 
here)  of  1167.2,  against  1167.6  as  shown  in  Table  1;  and  of  k,  .0537 
against  .0523.  Considering  that  the  constants  being  compared  are  de- 
rived from  different  groups  of  Guernsey  A.  R.  records,  they  are  in  sub- 
stantial agreement.  The  k's  are  directly  comparable,  but  the  A's  per- 
haps are  not,  since  Brody  seems  to  have  reckoned  t  at  the  end  of  the 
month,  whereas  it  is  reckoned  above  at  the  middle  of  the  month. 

Gestating  Cows. — The  milk  yields,  fat  yields,  and  fat-corrected 
milk  yields  for  the  twelve  groups  of  gestating  cows  are  given  in  Table  2. 
Equations  have  been  derived  only  for  the  F.C.M.  values  and  the  con- 
stants have  been  determined  graphically  by  Running's9  straight-line 
method,  Formula  V.  The  equations  are  given  in  Table  3.  The  calculated 
values  from  the  equations  and  deviations  of  the  observed  values  are  given 
in  Table  2. 

The  average  fat  percentage  is  also  given  in  Table  2.  The  fat 
percentage  measures  the  relative  rates  of  secretion  of  fat  and  of  milk 
as  a  whole,  and  also  affords  an  index  of  the  relative  rates  of  secretion  of 
the  several  milk  constituents.  This  significance  of  the  fat  percentage 
is  mentioned  to  justify  its  inclusion  with  data  on  the  rate  of  milk  secre- 
tion. , 

The  arrangement  of  Table  2  is  such  that  comparisons  may  be  made 
in  two  ways.  To  illustrate,  the  F.C.M.  yields  are  given  in  line  5  of  each 
group,  and  if  the  figures  in  line  5  for  any  group  are  followed  across  from 
left  to  right,  they  show  the  effect  of  advance  in  lactation,  and  after 
conception  the  combined  effect  of  advance  in  lactation  and  gestation, 
for  that  particular  group.  On  the  other  hand,  if  the  figures  of  lines  5  are 
read  from  top  to  bottom  of  any  column  they  show,  after  conception,  the 
effect  of  the  progress  of  pregnancy  with  the  stage  of  lactation  constant, 
but  they  involve  different  groups  of  cows.  The  heavy  zigzag  line  in  the 
table  indicates  where  conception  occurs,  that  is,  the  mid-point  of  the 
pregnancy  group  classification. 

The  use  of  different  groups  of  cows  introduces  an  uncertainty  on 
account  of  the  variability  in  yield.  If  one  reads,  for  example,  the  milk 
yields  in  the  first  month  of  lactation,  Table  2,  where  gestation  is  not  a 
factor,  it  is  seen  that  the  values  are  quite  irregular  from  group  to  group. 
On  this  account,  and  in  view  of  the  regularity  of  the  lactation  curve  that 
is  shown  by  the  farrow-cow  group,  it  seems  preferable  to  work  sepa- 
rately with  the  data  for  each  pregnancy  group. 

Calf  Carried  Five  Months  or  Less. — We  may  deal  with  the  first  six 
pregnancy  groups  more  or  less  collectively.  They  are  those  groups  where 


14 


BULLETIN  No.  272 


[January, 


TABLE  2. — AVERAGE  RATE  OF  MILK  SECRETION  PER  MONTH  BY  MONTHLY 
INTERVALS,  WITH  ADVANCE  IN  LACTATION  AND  GESTATION 


Advance  in 
gestation 

Line1 

Advance  in  lactation  —  month  after  calving  (mid-point) 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

Cows 
conceiving 
11.  5  months 
after 
calving 

I.. 

2.  .  . 
3... 
4.  .. 
5... 
6... 
7... 

162 
1102 
50  5 

196 
1046 

48  7 

196 
972 
46  7 

196 
918 
44  8 

196 
860 
43  7 

196 

824 
42  4 

196 
790 
40  8 

196 
752 
39  4 

194 
727 
38  1 

193 
692 
37  2 

193 
662 
36  1 

4  58 

4  66 

4.80 

4  88 

5  08 

5  15 

5  16 

5  24 

5  24 

5  38 

5  45 

1199 
1183 
16 

1150 
1137 
13 

1089 
1093 
-4 

1040 
1050 
-10 

1000 
1010 
-10 

966 
971 
-5 

927 
933 
-6 

892 
897 
-5 

863 
862 
1 

834 
829 
5 

804 
796 
8 

Cows 
conceiving 
10.5  months 
after 
calving 

1.. 
2... 
3... 
4... 
5... 
6... 
7... 

148 
1057 

48  7 

172 
1006 
46  5 

172 
937 
45  3 

172 
879 
43  3 

172 
833 
42  4 

172 
800 
41  0 

172 
767 
39  8 

172 
740 
38  7 

172 
704 
37  7 

172 
680 
36  4 

172 
642 
34  9 

4  61 

4  62 

4  83 

4  93 

5  09 

5  13 

5  19 

5.23 

5.36 

5  35 

5  44 

1153 
1134 
19 

1100 
1092 
8 

1054 
1052 
2 

1001 
1014 
-13 

969 
977 
-8 

935 
941 
-6 

904 
907 
-3 

877 
874 
3 

846 
842 
4 

818 
812 
6 

780 
782 
-2 

Cows 
conceiving 
9.5  months 
after 
calving 

1.. 
2... 
3... 
4.  .. 
5... 
6... 
7... 

188 
1135 
51.3 

235 
1079 
49.3 

235 
1009 
47.9 

235 
939 
45.9 

235 
882 
44  5 

234 
839 
43.0 

234 
797 
41.3 

234 
754 
39.5 

233 
725 
38.4 

233 
691 
37.1 

233 
660 
35.1 

4  52 

4  57 

4  75 

4  89 

5  05 

5  13 

5  18 

5  24 

5  30 

5.37 

5  32 

1223 
1218 
5 

1171 
1166 
5 

1123 
1117 
6 

1064 
1070 
-6 

1021 
1025 
-4 

980 
982 
-2 

938 
940 
-2 

895 
901 
-6 

866 
863 
3 

833 
826 
7 

791 
792 
-1 

Cows 
conceiving 
8.5  months 
after 
calving 

1.. 

2... 
3... 
4... 
5.  .. 
6... 
7... 

237 
1060 
50.4 
4.75 
1192 
1191 
1 

266 
1046 
48.8 
4.67 
1150 
1141 
9 

266 
973 
46.5 
4.78 
1087 
1093 
-6 

266 
905 
44.8 
4.95 
1034 
1048 
-14 

266 
857 
43.7 
5.10 
999 
1004 
-5 

266 
822 
42.5 
5.17 
966 
962 
4 

266 
780 
41.0 
5.26 
926 
922 
4 

266 
739 
39.3 
5.32 
885 
883 
2 

266 
705 
37.4 
5.30 
844 
846 
-2 

265 
672 
35.9 
5.34 
808 
811 
-3 

264 
642 
34.7 
5.40 

777 
777 
0 

Cows 
conceiving 
7.5  months 
after 
calving 

1.  . 
2.  .. 
3.  .. 
4... 
5.  .. 
6... 
7... 

312 
1067 
48.9 
4.58 
1160 
1165 
-5 

372 
1031 
47.6 
4.62 
1127 
1115 
12 

372 
959 
45.9 
4.79 
1072 
1068 
4 

372 
895 
43.8 
4.89 
1015 
1022 
-7 

372 
845 
42.5 
5.03 
976 
979 
-3 

372 
800 
41.1 
5.14 
937 
937 
0 

372 
764 
39.5 
5.17 
898 
897 

1 

372 
723 
37.7 
5.21 
854 
859 
-5 

372 
693 
36.3 
5.24 
821 
822 
-1 

372 
662 
35.0 
5.29 
790 
787 
3 

372 
634 
34.3 
5.41 
767 
753 
14 

Cows 
conceiving 
6.5  months 
after 
calving 

1.. 

2... 
3... 
4.  .. 
5... 
6... 
7.  .. 

487 
1089 
49.1 
4.51 
1173 
1173 
0 

575 
1042 
48.2 
4.63 
1140 
1125 
15 

575 
966 
46.1 
4.77 
1078 
1078 
0 

575 
904 
44.4 
4.91 
1028 
1033 
-5 

575 
854 
43.1 
5.05 
988 
990 
-2 

575 
814 
41.7 
5.12 
950 
949 
1 

575 
768 
39.8 
5.18 
905 
910 
-5 

575 
737 
38.2 
5.18 
868 
872 
-4 

574 
703 
37.1 
5.28 
837 
836 
1 

574 
667 
35.6 
5.34 
802 
802 
0 

571 
628 
34.4 
5.48 
768 
768 
0 

1Line  1,  number  of  records  averaged. 
Line  2,  raw  data  of  milk  yield,  in  pounds. 
Line  3,  raw  data  of  fat  yield,  in  pounds. 
Line  4,  average  fat  percentage. 

Line  5,  fat-corrected  milk  yield,  in  pounds,  observed. 
Line  6,  fat-corrected  milk  yield,  in  pounds  calculated. 
Line  7,  deviation  of  observed  from  calculated  F.C.M.  values,  in  pounds. 


effective  breeding  occurred  11.5,  10.5,  9.5,  8.5,  7.5,  and  6.5  months 
after  calving.  The  data  are  given  in  the  first  part  of  Table  2,  and  are 
shown  graphically  in  Figs.  2  to  7.  By  study  of  the  graphs,  particularly, 
it  is  evident  that  equation  (7)  fits  the  observations  very  well,  apparently 
just  as  well  as  in  the  case  of  the  farrow  cows.  It  appears,  therefore,  that 
for  the  first  five  months  of  the  gestation  period  pregnancy  does  not 
appreciably  affect  the  lactation  curve. 

Calf  Carried  More  Than  Five  Months.— After  the  fetus  has  reached 
an  age  of  five  months,  the  lactation  curve  is  modified  more  or  less,  and 


1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION 


15 


TABLE  2. — Concluded 


Advance  in 
gestation 

Line 

Advance  in  lactation  —  month  after  calving  (mid-point) 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

Cows 
conceiving 
5.5  months 
after 
calving 

1.. 
2... 
3... 
4.  .. 
5.  .. 
6... 
7... 

636 
1081 
49  2 

754 
1037 
47.8 

753 
967 
46.0 

753 
907 
44.3 

753 
851 
42.4 

753 
809 
40.8 

753 
771 
38.9 

752 
734 
37.8 

752 
698 
36.4 

752 
660 
35.1 

747 
611 
33.4 

4.55 
1170 
1174 
-4 

4.61 
1132 
1123 
9 

4.76 
1077 
1074 
3 

4.88 
1028 
1027 
1 

7.42 
977 
982 
-5 

5.04 
936 
939 
-3 

5.05 
893 
898 
-5 

5.15 
860 
859 
1 

5.22 
825 
821 
4 

5.32 
791 

784 
7 

5.47 
746 
746 
0 

Cows 
conceiving 
4.5  months 
after 
calving 

1... 
2... 
3... 
4... 
5... 
6... 
7... 

615 
1064 
48  2 

746 
1018 

47.4 

746 
949 
45.4 

746 
883 
43.3 

745 
827 
41.2 

745 
784 
39.5 

745 
745 
38.1 

745 
707 
36.9 

745 
660 
35.3 

742 
609 
33.3 

727 
539 
30.8 

4.53 
1148 
1163 
-15 

4.66 
1118 
1108 
10 

4.78 
1061 
1055 
6 

4.90 
1003 
1005 
-2 

4.98 
949 
957 

-8 

5.04 
906 
910 
-4 

5.11 

869 
868 
1 

5.22 
836 
825 
11 

5.35 
793 

784 
9 

5.47 
744 
740 
4 

5.68 
675 
684 
-9 

Cows 
conceiving 
3.5  months 
after 
calving 

I... 

2... 
3.  .. 
4.  .. 
5.  .. 
6... 
7... 

439 
1075 
49.3 
4  59 

555 
1021 
48.0 
4  70 

555 
948 
45.8 
4.83 

555 
889 
43.5 
4  89 

555 
832 
41.1 
4.94 

555 
791 
39.9 
5  04 

554 
744 
38.3 
5  15 

554 
697 
36.7 
5  27 

553 
643 
34.8 
5  41 

542 
570 
31.7 
5  56 

486 
502 
28.8 
5  74 

1169 
1178 
-9 

1128 
1119 
9 

1067 
1063 
4 

1008 
1010 
-2 

950 
960 
-10 

914 
912 
2 

872 
865 

7 

829 
820 
9 

779 
774 
5 

704 
719 
-15 

633 
633 
0 

Cows 
conceiving 
2.5  months 
after 
calving 

I.. 

2... 
3.  .. 

4.  .. 
5... 
6... 

7... 

183 
1110 
51  0 

235 
1068 
49  6 

235 

987 
46.7 

235 
915 
44  2 

235 

867 
42  6 

235 
813 
41  6 

235 
769 
40  0 

235 
724 
38  5 

235 
664 
36  4 

235 
580 
33  1 

235 
478 
28  1 

4.59 
1209 
1203 
6 

4.64 
1171 
1148 
23 

4.73 
1095 
1094 
1 

4.83 
1029 
1044 
-15 

4.91 
986 
995 
-9 

5.12 
949 
948 
1 

5.20 
908 
903 
5 

5.32 
867 
858 
9 

5.48 
812 
808 
4 

5.71 
729 
739 
-10 

5.88 
613 
612 
1 

Cows 
conceiving 
1.5  months 
after 
calving 

1.. 
2... 
3.  .. 

4... 
5... 
6... 
7.  .. 

104 
1120 
50.5 
4.51 
1206 
1192 
14 

132 
1052 
48.5 
4.61 
1148 
1136 
12 

132 
969 
45.6 
4.71 
1072 
1083 
-11 

132 
908 
43.7 
4.81 
1019 
1032 
-13 

132 
852 
41.9 
4.92 
969 
983 
-14 

132 
812 
41.0 
5.05 
940 
936 
4 

132 

768 
39.8 
5.18 
904 
889 
15 

132 
702 
37.4 
5.33 
842 
837 
5 

132 
614 
33.9 
5.52 
754 
766 
-12 

132 
508 
28.9 
5.69 
637 
637 
0 

Cows 
conceiving 
.5  months 
after 
calving 

1... 
2... 
3... 
4.  .  . 
5.  .. 
6... 
7... 

23 
902 
41.8 
4.63 
988 
1061 
-73 

29 
940 
43.2 
4.60 
1024 
1014 
10 

29 
881 
40.4 
4.59 
958 
970 
-12 

29 
812 
39.6 
4.88 
919 
926 
-7 

29 

778 
37.7 
4.85 
877 
884 
-7 

29 
750 
37.3 
4.97 
860 
842 
18 

29 
664 
34.6 
5.21 
785 
795 
-10 

29 
598 
32.7 
5.47 
730 
728 
2 

29 
481 
27.3 
5.68 
603 
603 
0 

NOTE. — The  heavy  zigzag  line  thru  the  table  denotes  approximately  the  time  of  conception.  Data 
from  left  to  right  show  the  effect  of  advance  in  lactation,  plus  (to  the  right  of  the  heavy  line)  the  effect 
of  advance  in  gestation  for  the  same  group  of  cows.  Data  from  top  to  bottom  show  the  effect  of  advance 
in  gestation  (below  the  heavy  line)  independent  of  stage  of  lactation,  but  involve  different  groups  of 
cows. 

equation  (10)  is  used  in  place  of  (7).  The  data  are  given  in  Tables  2  and 
3,  and  shown  graphically  in  Figs.  8  to  13.  The  groups  represented  in 
Figs.  11,  12,  and  13  have  carried  the  calf  8.5  months  at  the  last 
observation  and  show  a  more  extended  effect  of  pregnancy  on  the  lacta- 
tion curve  than  occurs  in  Figs.  8,  9,  and  10. 

It  will  be  noticed  from  Table  3  that  the  constant  .K  is  the  same  in  each 
of  the  six  groups  where  equation  (10)  is  used.  The  constant  B  is  also  the 
same  in  four  of  the  six  groups.  This  might  be  taken  to  indicate  that  the  de- 
crease in  yield  due  to  pregnancy  is  a  fixed  quantity,  measured  by  the  term 
BeK(i~c)(B  =  .01206,  K  =  1.09861).  Representing  this  decrease  by  i 

di 

(inhibition)  and  time  in  months  of  pregnancy  by  p,  we  have  —  =  beKp, 

dp 


16 


BULLETIN  No.  272 


[January, 


Fig  Z  -  Bat*  Of  MilK.   Secret 
(»«ar/ts  time  of  concrpikm) 


T/me  In  Months  From  Calving  (t) 


9  10  II 


•c   1000 
I 


-  Sate  Of  Mi'lK  Secretion 
+  MarKi   time  of  conception} 


Time  In  Months  From 


i)      800 

<x. 

*: 


Time  In  Months  from  Calvirg  (t) 


representing  the  rate  of  decrease  in  yield  due  to  pregnancy  (b  =  .01147). 
The  gestation  period  is  9.2  months  and  the  decrease  in  yield  for  the 

entire  gestation  period  would  be    I     beKp  dp  =  ^  (e9'2   —    1)    =  256. 

Jo  K. 

This  result  is  in  terms  of  fat-corrected  milk  in  pounds.  Converting 
to  an  energy  basis  as  per  the  relations  mentioned  on  pages  6  and  7 
gives  85  therms  (1  therm  =  1,000  large  calories). 


1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION 


17 


.g 
C    iz 


Fig  S-  Hate  Of  M,l<  Secretion 


Time.  In  Honihs  Tram  Celring  (t) 


•S    woo 


Tig  6-  Kate  Of  MM  Secretion 
(  #  Marls  time  of  conception) 


Tim*  In  Months  fro 


Time  In   Months  from  Caln'ng  (t) 


The  decrease  in  yield  for  the  gestation  period  of  the  group  bred 
4.5  months  after  calving  (Fig.  9)  would  be,  according  to  the  equation, 
nearly  twice  as  great  as  the  above  result.  This  equation  is  not  so  reliable 
as  the  equations  of  Figs.  11,  12,  and  13,  because  the  observations  on 
which  it  is  based  do  not  extend  past  6.5  months  of  pregnancy,  that  is, 
into  that  portion  of  the  gestation  period  where  the  effect  on  yield  is 
most  appreciable. 


18 


BULLETIN  No.  272 


[January, 


<:    IOOO 
I 


Fia  3-  Bate  Of  Mi  IK.  Secretion 
(  +  M<irKs    time,  of  conception) 


Time.  In  Mont  hi    from   Calling  (tj 


•&>  IZC 


9-  Kate  Of  MM  Secret  ion 
MarKl  time  of  conception) 


77m«  In  Months  From  Calving  (t) 


i&  10-  Rat*  Of  Him  Secreti 
(  •  Marf>  iimt  of  conceftio 


Time.  In  Months  From  Catr'ng(tJ 


It  seems  desirable,  however,  to  study  somewhat  further  the  con- 
stancy of  the  decrease  in  yield  which  is  associated  with  pregnancy.  It 
may  be  that  the  effect  varies  with  the  age  or  productive  level  of  the  cow. 

Age  and  Productive  Level. — The  groups  bred  at  1.5  and  2.5  months 
after  calving  have  each  been  divided  into  four  age  groups:  (1)  one  and 
two  years,  (2)  three  and  four  years,  (3)  five  and  six  years,  and  (4)  seven 
years  and  over.  The  yields  have  been  computed  as  before  and  equation 


1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION 


19 


f- 
I 


Fig  II -Gate  Of  ft>/K  Jfcretion 
(  »  narki  iima  of  conception) 


T;m«   In   Month*   fr 


CalYinj)(t) 


Time.  In  Month)    Trom  Celyin£({) 


i£  /3-Pd/e    Of  Mi'lK  Jecret, 
(  »  Marti  </m«  of  conception) 


T,me    In    Month)    from   CaMng(t) 


(10)  applied  to  the  F.C.M.  values.  The  data  of  yields  are  given  in 
Tables  4  and  5,  the  equations  in  Table  6,  and  graphic  presentation  in 
Figs.  14  to  21.  From  the  deviations  recorded  in  Tables  4  and  5,  and 
from  Figs.  14  to  21,  it  will  be  seen  that  on  the  whole  the  smooth  curve 
fits  the  data  satisfactorily.  From  Table  6  it  may  be  noted  that  the  con- 
stant K,  representing  the  rate  of  change  in  the  rate  of  inhibition  of  milk 
secretion,  has  the  same  value  as  before,  1.09861,  except  in  one  case. 


20 


BULLETIN  No.  272 


[January, 


i'g  It-  Rtt*  Of  MilK  Sfcretn 
Cows  Age  I  and  Z  Yfiri 

(*  Mart!)   time  of  conception} 


Time    In  Month)  From  Calr~mg(t) 


Fig  /6-  Sate  Of  MilK  Secret  ic, 
Cow*  Age  3 and  4  tear) 

time  of  conception) 


Time.  In  Month)    From  Cjlrmgft) 


ftf  If,-  Kite  Of  Mil*  *ecret,o 

Co*~->  Age  5jnd6  /«/-> 
(•  M<irX>  time  of  conception) 


Time  In  Month)  from  Cjlr,ng(t) 


f,$-n-Kate  Of  M,IK  Secretior 
Co»>  A/je  7anJ  «•  XM/-J 

time  of  conception) 


Tine  In  Monthi  fro 


1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION  21 


T 1 1 r 


f/^  IS-  Sate  Of  MilK  Jffcretioi 

Cowi  Age  I  and  2  Yeirt 
(IMirKs  time  of  concept: on) 


Time.  In  Month)  Tr 


Fij  19-  Sate  Of  MM  Secret, 01 

Cowi  Age  3 and  4  X<-dr3 
(*  Marts  time  of  conception) 


Tine.  In  Monthi  Tr 


10-  Bite  OfM'iK  Secretion 
Cows  AgeSand6yrar3 
ftarKi  time  of  conception) 


.  In  Monthi    fro 


fig  Zl-  Rita  Of  MilK  Secretio 
COM)  t&t  7  end  *  Years 

time,  of  conception) 


Tim*  In    Month*     from    Ctlrin4(t) 


BULLETIN  No.  272 


[January, 


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1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION 


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1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION 


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26 


BULLETIN  No.  272 


[January, 


The  values  of  B,  however,  show  considerable  variation  between  the 
different  age  classes.  The  value  of  B  seems  to  be  more  closely  related 
to  the  productive  level,  as  represented  by  A,  than  it  is  to  the  age  of  the 
cows  composing  the  group. 

It  may  be  noted  also  from  Table  6  that  the  value  of  k  tends  to  in- 
crease with  age  and  yield;  that  is,  the  younger  cows  are  more  "per- 
sistent" than  the  older  cows.  Just  how  far  this  greater  persistency  is 
associated  with  younger  age  independent  of  yield,  and  how  far  with  a 
lower  absolute  production,  independent  of  age,  the  present  treatment 
of  the  data  does  not  distinguish. 

Fat  Percentage. — The  change  which  takes  place  in  fat  percentage 
with  advance  in  gestation  independent  of  advance  in  lactation  may  be 
determined  from  Table  2  by  reading  vertically,  that  is,  from  one  group 
to  another.  As  between  the  several  groups  at  the  same  stage  of  lactation 
and  before  breeding,  there  is  very  little  difference  in  the  average  fat 
percentage.  Direct  reading  from  the  table  then  offers  a  simple  and 
satisfactory  way  of  studying  the  changes  in  fat  percentage  associated 
with  gestation.  The  data  from  columns  9,  10,  and  11  of  Table  2,  repre- 
senting months,  the  mid-points  of  which  are  respectively  9,  10,  and  11 
months  from  calving,  are  represented  in  Fig.  22.  There  is  an  increasing 


Fig  22-  Effect  of  Pregnancy  on  fat  Percentage 
Stage  of  Lactation  Constant 


•S  IS  2.5  3.S  -4.5  5.5  *.S  J.S  8-5  9.5 

Time   From  Conception  — Month*  (Mid-Vomti  of  Months) 

tendency  for  the  fat  percentage  to  rise  with  advance  in  gestation  after 
the  fetus  reaches  four  months  of  age.  There  is  no  particular  object  in 
attempting  to  find  an  equation  for  this  rise,  but  it  appears  to  be  an  ex- 
ponential change.  The  fat  percentage  appears  to  be  appreciably  affected 
at  a  slightly  earlier  stage  of  gestation  than  is  the  case  with  yield.  Here 
also,  as  seems  to  be  quite  generally  the  rule,  when  the  rate  of  milk 
secretion  as  a  whole  is  changing  appreciably,  the  change  in  the  rate  of 
fat  secretion  tends  to  lag  behind  the  change  in  the  rate  at  which  the  other 
constituents,  taken  collectively,  are  secreted. 


1926}      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION  27 

DISCUSSION 

From  the  numerical  and  graphic  presentation  preceding,  it  is  evi- 
dent that  the  lactation  curve,  as  represented  by  the  average  of  Guernsey 
Advanced  Registry  records  of  farrow  cows,  and  also  of  gestating  cows 
up  to  the  end  of  the  fifth  month  of  pregnancy,  is  very  well  represented 

dii 

by  the  differential  equation  (1),  --    =  ae~kt,  notation  as  given  above 

at 

(page  7).  In  biological  material  as  variable  as  milk  yield,  one  would 
hardly  expect  to  find  better  agreement  than  is  shown  between  the 
observed  monthly  yields  and  the  calculated  values.  The  root-mean- 
square  error  of  the  observations  for  farrow  cows  (Table  1),  is  less  than 
one-half  percent  of  the  average  monthly  yield.  The  maximum  deviation 
at  any  observation  after  the  first  for  gestating  cows  is  18  pounds  from  the 
calculated  value  of  842  pounds  (Table  2).  The  larger  deviations  occur, 
as  a  rule,  in  the  observations  of  the  earlier  part  of  the  lactation. 

The  two  constants  of  the  equation  can  be  given  definite  significance 
in  terms  of  commonly  recognized  characters.  The  constant  a,  represent- 
ing the  initial  rate  of  secretion  per  month,  is  descriptive  of  the  pro- 
ductive ability  of  the  cow  at  the  first  of  lactation,  that  is,  for  a  month 
period.  On  the  other  hand,  the  constant  k,  considered  as  a  positive 
quantity,  representing  the  rate  of  decrease  per  month  in  the  rate  of 
yield  per  month,  is  descriptive  of  that  character  commonly  known  as 
persistency.  It  is  an  inverse  measure  of  persistency  as  the  term  is  gen- 
erally known. 

The  theoretical  yield  for  the  lactation  period  or  any  portion  of  it  is 
a  function  of  a  and  k  For  the  entire  lactation  the  theoretical  yield  of 
farrow  cows  is  a/k.  For  a  fixed  value  of  k  the  yield  for  the  lactation  or 
any  portion  of  it  is  proportional  to  a.  For  a  fixed  value  of  a  the  yield 
for  the  lactation  is  inversely  proportional  to  k ;  but  for  a  partial  lactation 
the  relation  is  inverse  but  not  strictly  proportional. 

Brody,  Ragsdale,  and  Turner  have  pointed  out  particularly  that 
the  form  of  the  equation  for  the  lactation  curve  is  the  form  that  also  de- 
scribes the  course  of  a  monomolecular  reaction;  and  they  infer  that  the 
rate  of  milk  secretion  is  controlled  by  some  such  chemical  process.  It 
seems  clear  from  what  we  know  of  the  nutrition  of  the  mammary  gland 
that,  in  the  main,  the  elemental  materials  and  energy  required  in  the 
elaboration  of  milk  in  the  gland  are  supplied  more  or  less  continuously 
from  the  food  thru  the  blood.  The  assumption  of  the  presence  of  some 
material  in  the  nature  of  a  catalyst  which  acts  as  a  necessary  inter- 
mediary in  the  chemical  processes  involved,  is  perhaps  reasonable,  as  is 
also  the  assumption  that  the  catalyst  is  very  slowly  and  continuously 
destroyed  in  accordance  with  the  equation.  It  would  seem  to  be  necessary 


28  BULLETIN  No.  272  [January, 

to  assume  that  the  hypothetical  catalyst  is  confined  to  the  milk-secreting 
cells,  since,  so  far  as  the  writers  are  aware,  attempts  to  show  the  presence 
of  an  accelerating  agent  to  milk  secretion  in  the  circulation  of  actively 
lactating  animals  have  given  negative  results. 

Another  possible  explanation  of  the  form  of  the  lactation  curve 
occurs  in  connection  with  the  known  calcium  requirements  of  the 
lactating  cow.  The  calcium  outgo  exceeds  the  calcium  intake  while 
lactation  proceeds  above  a  certain  level.  The  depletion  of  the  calcium 
reserves  of  the  body  might  conceivably  lead  to  a  reduction  in  the  rate 
of  secretion  in  accordance  with  that  found.  At  low  yields  calcium  would 
not  seem  to  be  a  possible  limiting  factor,  but  it  is  not  certain  that  the 
rate  of  secretion  continues  to  decline  in  conformity  with  the  equation 
after  a  low  value  is  reached. 

Equation  (9),  •-  =  ae~kt  -  beK(t~c},  (page  10),    as  applied  thru 
at 

equation  (10),  serves  to  describe  satisfactorily  the  lactation  curve  for 
cows  in  advanced  gestation,  as  evidenced  by  the  deviations  given  in 
Tables  2,  4,  and  5  and  by  the  graphic  presentation  in  Figs.  8  to  21. 
The  minus  term  of  equation  (9)  represents  the  effect  of  pregnancy  on 
yield.  The  values  found  for  the  constants  (6  =  .01147;  K  =  1.09861) 
give  a  result  which  is  in  accord  with  results  of  other  investigators, 
namely,  that  pregnancy  results  in  a  constantly  increasing  reduction  of 
yield,  which,  however,  is  scarcely  appreciable  during  the  first  five 
months.  The  equation  gives  a  total  decrease  for  the  first  five  months 
of  2.5  pounds,  but  the  decrease  for  the  entire  period  (9.2  months)  is  256 
pounds.  The  rate  of  decrease  as  represented  by  the  equation  is  shown 
graphically  in  Fig.  23. 

Gowen,11  by  correlation  methods  applied  to  the  records  of  Vol.  31 
of  the  Guernsey  Advanced  Register,  found  a  decrease  in  milk  yield, 
due  to  carrying  the  calf  9  months,  of  342  pounds  in  cows  3  to  3%  years 
old;  and  of  628  pounds  in  cows  3^  to  4  years  old.  Assuming  the  milk 
to  contain  5  percent  fat,  this  would  give  F.C.M.  values  of  393  and  722 
pounds  respectively.  Gowen's  results  are  thus  considerably  higher  than 
found  above.  His  results  are  based  on  a  linear  relation  between  duration 
of  pregnancy  and  reduction  of  milk  yield.  The  present  results  indicate 
that  the  relation  is  distinctly  non-linear. 

With  reference  to  effect  of  pregnancy  on  composition  of  the  milk, 
as  indicated  by  the  fat  percentage,  Gowen  concludes  that  there  is  no 
influence.  His  conclusion  is  based  on  the  correlation  coefficient  for  fat 
percentage  for  the  year  and  duration  of  pregnancy.  The  present  results, 
however,  show  a  quite  pronounced  increase  in  fat  percentage  accom- 
panying advance  in  gestation,  independent  of  advance  in  lactation 


1926]      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION  29 

(Fig.  22).  This  increase  would,  of  course,  be  much  less  pronounced  if 
merged  in  the  average  data  for  a  year. 

The  differences  in  the  present  results  and  those  of  Gowen  must  be 
due  to  the  differences  in  method  of  treating  the  data,  since  the  source 
of  the  data  in  both  cases  is  essentially  similar.  A  reason  for  the  difference 
is  not  hard  to  find.  As  to  the  absolute  amount  of  the  decrease  in  yield 
associated  with  pregnancy,  the  partial  correlation  method  used  by  Gowen 
takes  account  of  the  difference  in  level  of  production  of  the  farrow  cows 
and  pregnant  cows  at  the  third  month  of  lactation.  It  fails  to  take 
account  of  the  difference  between  the  several  groups  in  rate  of  decline 
in  yield  as  lactation  advances.  From  Table  3  it  may  be  observed  that 
the  pregnant  cows  tend  to  have  a  higher  value  of  the  k  constant,  that 
is,  they  decline  more  rapidly  in  yield  than  the  farrow  cows.  There  ap- 
pears to  be  no  reason  to  attribute  this  difference  to  pregnancy.  A  further 
limitation  of  the  correlation  method  is  that  its  accuracy  depends  upon 
linearity  of  the  regression,  and  this  condition  is  not  fully  satisfied  in 
the  present  case. 

As  to  the  increase  in  fat  percentage  of  the  milk  associated  with 
advanced  pregnancy  shown  in  Fig.  22,  a  rough  estimate  indicates  that 
the  average  fat  percentage  for  the  year  would  be  increased  by  less  than 
.1  in  cows  carrying  the  calf  9  months.  For  cows  carrying  the  calf  5 
months,  the  increase  in  the  yearly  fat  percentage  is  negligible.  Hence 
the  coefficient  of  correlation  between  duration  of  pregnancy  and  average 
fat  percentage  for  the  year  would  be  very  low.  Gowen  found  the 
coefficient  to  be  .014  as  an  average  of  ten  age  classes.  But  it  is  not  safe 
to  conclude  from  this  low  coefficient  that  the  two  variables  are  not  at  all 
associated.  The  correlation  coefficient  as  applied  is  simply  inadequate 
to  show  the  relation  in  sufficient  detail  in  this  particular  case. 

In  explanation  of  the  decrease  in  yield  with  pregnancy  two  sug- 
gestions have  been  offered  at  various  times — one,  that  the  decrease  is 
due  to  the  nutrients  required  by  the  fetus;  the  other  that  a  hormone  is 
produced  and  enters  the  circulation  during  pregnancy  and  acts  as  an 
inhibitor  to  milk  secretion. 

Experimental  evidence  indicates  that  the  net  nutrients  require- 
ment for  gestation  constitutes  a  very  minor  drain  on  the  mother.  Thus, 
Eckles,12  as  the  result  of  careful  experiments,  found  that  pregnant  cows 
on  a  maintenance  ration  for  farrow  cows  finished  gestation  with  fully 
nourished  calves  and  with  their  own  body  weights  unimpaired. 

If  the  decrease  in  milk  yield  during  the  course  of  pregnancy  is  due 
to  the  nutrients  requirement  of  the  fetus,  then  the  curve  of  the  rate  of 
decrease  might  be  expected  to  parallel  the  growth-weight  curve  of  the 
fetus.  Unfortunately  accurate  data  on  the  growth  of  the  bovine  fetus 


30 


BULLETIN  No.  272 


[January, 


are  not  available.  The  equation  of  the  first  extra-uterine  growth  cycle 
of  the  Jersey  female  has  been  determined  by  Brody  and  Ragsdale.13 
There  is  no  reason  to  suppose  that  the  course  of  growth  of  the  calf  in 
the  late  stages  of  intra-uterine  development  should  differ  greatly  from 
that  immediately  following  birth.  Robertson,14  in  fact,  has  presented 
indirect  evidence  from  the  data  of  Brody  and  Ragsdale  that  their 
equation  represents  also  the  late  stages  of  intra-uterine  growth.  Robert- 
son has  shown  also  that  a  similar  relation  exists  in  the  growth  of  the 
infant.  The  data  of  Donaldson21  and  of  Read22  on  the  intra-uterine 
growth  of  the  rat  and  the  guinea  pig,  tend  also  to  support  the  growth 
relations  indicated  above.  For  lack  of  better  data  we  may  use  the  ex- 
trapolated values  of  Brody  and  Ragsdale's  equation  in  comparing  the 
decrease  in  milk  yield  with  the  growth  of  the  fetus. 


of  Inhibition 
of  retui  X5 


Time   In  Month)  from   Conception 

FIG.  23. — SHOWING  THE  OBSERVED  RATE  OF  INHIBITION  AND  THE 
PROBABLE  WEIGHT  OF  THE  FETUS  WITH  ADVANCE  IN  GESTATION 

The  solid-line  curve  shows  the  rate  of  inhibition  per  month 
in  the  rate  of  yield  per  month  with  advance  in  gestation.  The  area 
under  this  curve  represents  the  decrease  in  yield  up  to  nine  months 
due  to  pregnancy.  The  broken-line  curve  represents  the  probable 
weight  of  the  fetus.  The  rate  of  inhibition  does  not  parallel  the 
rate  of  the  probable  draft  of  the  fetus  on  the  nutrients  of  the  ma- 
ternal blood  stream. 

It  would  seem  that  if  the  nutrients  requirement  of  the  fetus  is  the 
cause  of  the  decrease  in  milk  yield,  the  rate  of  decrease  should  be  pro- 
portional to  the  weight  of  the  fetus,  on  the  assumption  that  the  re- 
quirements of  the  fetus  at  any  moment  are  proportional  to  its  weight. 
This  appears  to  be  true  in  the  case  of  the  chick  (cf.  Needham,23  Figs. 
6  and  8).  The  two  curves  are  given,  therefore,  in  Fig.  23,  the  one  to 
represent  the  weight  of  the  fetus,  the  other  to  represent  the  rate  of  de- 
crease in  yield  due  to  pregnancy.  The  two  curves  are  not  parallel,  and 


1926]      MILK  SECRETION  WITH  ADVANCE  IN  LATCATION  END  GESTATION  31 

it  seems  unlikely  that  the  rate  of  milk  secretion  is  affected  at  all  pro- 
portionately to  the  nutrients  required  by  the  fetus. 

Lane-Claypon  and  Starling15  found  experimental  evidence  of  a 
hormone  of  pregnancy  which  induces  growth  of  the  mammary  gland 
and  they  suggested  that  the  hormone  acted  also  as  an  inhibitor  to 
milk  secretion.  D'Errico16  supports  this  inhibitor  postulate  on  the 
basis  of  the  results  of  blood  transfusions  from  a  pregnant  to  a  lactat- 
ing  bitch,  showing  a  transitory  decrease  in  the  rate  of  milk  secretion. 

Woodman  and  Hammond17  report  that  the  amount  and  character 
of  the  secretion  of  the  mammary  gland  of  heifers  in  first  pregnancy 
undergoes  a  noticeable  change  in  the  fifth  month.  This  is  also  the  time 
at  which  the  effect  of  pregnancy  on  milk  secretion  becomes  apparent. 

A  hormone  of  pregnancy  inhibiting  milk  secretion  and  appearing 
about  the  fifth  month  seems  to  offer  a  plausible  explanation  of  the 
decrease  in  rate  of  milk  secretion  in  late  gestation,  and  has  been  accepted 
by  various  investigators  (cf .  Hammond  and  Sanders,2  Eckles18  page  413, 
and  Hooper19).  The  rate  at  which  such  an  inhibitor  may  be  produced 
need  not  be  proportional  to  the  size  of  the  fetus. 

Some  rather  meager  data  by  Gaines20  indicate  that  there  is  not 
only  an  inhibitor  to  milk  secretion  present  in  the  blood  of  the  pregnant 
goat,  but  that  it  persists  for  some  time  after  parturition  in  the  circulation 
of  both  mother  and  kid.  This,  together  with  results  mentioned  above, 
suggests  the  influence  of  pregnancy  on  the  mammary  gland  presented 
diagrammatically  in  Fig.  24.  The  figure  represents  lactation  as  con- 
tinuous, that  is,  as  without  any  dry  period,  and  covers  the  last  2.5 
months  of  gestation  and  the  first  1.5  months  of  the  lactation  following. 

Teleologically  one  might  say  that  gestation  provides  a  mechanism 
which  insures  the  development  and  preparation  of  the  mammary  gland 
for  the  secretion  of  milk  for  the  post-natal  nourishment  of  the  young, 
and  which  inhibits  the  secretion,  almost  or  entirely,  preceding  par- 
turition, and  by  the  gradual  removal  of  the  inhibitor  following  partu- 
rition provides  for  some  time  an  increasing  milk  supply  to  meet  the 
increasing  needs  of  the  growing  young. 

If  it  be  assumed  that  the  production  of  the  inhibitor  ceases  at 
parturition  and  that  the  amount  then  present  in  the  circulation  is 
destroyed  or  eliminated  at  a  rate  proportional  to  its  concentration  at 
the  moment,  then  the  rate  of  milk  secretion  following  parturition  would  be 

fltJ 

represented  by  the  general  equation  —   =  ae~kt  —  be~klt,  in  which  the 

last  term  represents  the  post-partum  inhibition  of  pregnancy.  Brody, 
Turner,  and  Ragsdale7  have  applied  this  equation,  as  representing  the 
course  of  a  consecutive  chemical  reaction,  to  observed  milk  yields 


32 


BOLLETIN  No.  272 


[January, 


immediately  following  parturition,  and  secured  a  very  satisfactory  fit. 
They  have  accordingly  interpreted  the  rate  of  milk  secretion  for  the 
first  month  or  two  of  lactation  as  dependent  on  such  a  reaction  or  pro- 
cess. An  alternative  interpretation  is  suggested  by  the  considerations 
presented  in  Fig.  24. 


5    eoo 


Time  In  Months  from 
FIG.  24. — INFLUENCE  OF  PREGNANCY  ON  THE  RATE  OF  MILK  SECRETION 

Profound  changes  in  the  activity  of  the  mammary  gland  occur 
during  late  pregnancy  and  following  parturition.  The  lines  F  repre- 
sent projections  of  the  lactation  curve  as  found  outside  the  influence 
of  pregnancy,  parturition  recurring  before  lactation  ceases.  The  hor- 
izontally shaded  area  represents  the  effect  of  pregnancy  in  promoting 
the  growth  or  rejuvenescence  of  the  gland  with  reference  to  its  po- 
tential secreting  capacity;  this  effect  is  pictured  as  abruptly  termin- 
ated at  parturition.  The  vertically  shaded  area  represents  the  decrease 
in  milk  yield  observed  in  gestating  cows  as  compared  with  the  ex- 
pected yield  when  unbred.  The  horizontally  and  vertically  shaded 
areas  together  represent  the  inhibition  of  milk  secretion  preceding 
parturition.  The  persistence  of  this  inhibition  after  parturition  is  rep- 
resented by  the  stipled  area.  The  ordinates  of  the  lower  border  of  the 
shaded  areas  represent  the  rate  of  milk  secretion.  The  ordinates  of 
the  upper  border  of  the  shaded  areas  represent  the  potential  capacity 
of  the  gland  as  to  rate  of  secretion.  The  shaded  portions  of  the 
ordinates  represent  the  rate  of  inhibition  of  the  rate  of  milk  secretion 

It  is  assumed  in  Fig.  24  that  the  hormone  which  inhibits  milk 
secretion  is  separate  from  that  which  promotes  growth  and  rejuven- 
escence of  the  gland.  If  the  two  are  the  same  or  if  the  latter  also  persists 
after  parturition,  it  becomes  necessary  to  modify  somewhat  the  upper 
curve  of  Fig.  24  to  take  account  of  the  influence  after  parturition.  Under 
these  conditions  the  sharp  point  of  the  curve  would  be  rounded  down. 
But  since  this  modification  is  small  in  value  as  compared  to  the  total 


1926]      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION  33 

values  where  it  applies,  it  would  not  materially  affect  the  applicability 
of  the  above  equation. 

CORRECTION  FACTORS  FOR  LENGTH  OF  RECORD 

The  lactation  curve,  that  is  the  rate  of  milk  secretion,  may  be 

dy 

expressed  as   —  =  ae~kt  except  as  to  certain  complications  associated 
at 

with  pregnancy.  Outside  the  influence  of  pregnancy,  therefore,  the 
equation  of  the  lactation  curve  offers  a  basis  for  determining  the  quanti- 
tative relations  between  the  yields  for  various  periods  of  the  lactation. 
If  we  represent  the  standard  length  of  record  by  S  and  wish  to  secure 
the  ratio  of  S  to  some  other  length  of  record,  L,  both  records  starting 
at  the  same  time,  n,  after  calving,  the  ratio  is: 

'n  +  S 


/" 

Jn 


ae~ktdt      , 

1  —  e~sk 


n+L         ... 

ae~ktdt 

From  Table  3  the  average  value  of  A;  is  .04412.  If  time  is  reckoned 
in  days  instead  of  months,  k  becomes  .04412/30.5  =  .00144656.  By  using 
this  value  of  k  and  305  days  as  the  standard  length  of  record,  the  values 
shown  in  Table  7  are  obtained  from  the  above  ratio.  It  should  be  borne 
in  mind  that  the  factors  given  in  Table  7  are  based  on  a  constant  value 
of  k  and  their  applicability  is  limited  by  this  fact.  According  to  the  table, 
the  305-day  yield  is  86.97  percent  of  the  365-day  yield;  but  if  k  becomes 
.1  (instead  of  .04412)  the  305-day  yield  becomes  90.46  percent  of  the 
365-day.  Between  individuals  there  is  presumably  some  variability  in 
the  values  of  k.  From  Table  6  it  seems  that  k  varies  also  with  age  and 
productive  level.  The  factors  of  Table  7  are  offered,  therefore,  tenta- 
tively and  as  an  approximation. 

CORRECTION  FACTORS  FOR  PREGNANCY 

The  correction  factors  for  pregnancy  are  derived  from  the  equations 

dii 

of  the  lactation  curves  for  farrow  and  gestating  cows:  -r-   =    ae~kl  and 

at 

dy  K    _  . 

—    =  ae~kt  —  be  ,  respectively.    Using  the  record  of  farrow  cows 

for  a  period  of  ten  months  as  the  standard,  the  ratio  of  that  record  to  the 
corresponding  record  of  gestating  cows  is  given  by: 

—  M     —    a-lOk\  J?  f-l  a-Wh\ 

!„  U       e       )  J\.  (L  —  e       ) 


_ 


34 


BULLETIN  No.  272 


[January, 


Since  c  =  months  from  calving  to  conception,  10-c  =  months  of  preg- 
nancy. From  values  previously  given,  K  =  1.09861  and  k  =  .04412. 
From  Table  6,  b/a  =  .00001,  approximately.  By  applying  these  values 
in  the  above  ratio,  the  correction  factors  for  pregnancy  given  in  Table 
8  have  been  derived. 

Since  the  correction  for  pregnancy  is  of  small  magnitude,  it  is  per- 
missible first  to  correct  the  record  for  length  of  time,  if  different  from 
305  days,  and  then  to  apply  the  correction  of  Table  8  to  take  care  of 
the  time  the  calf  is  carried. 


TABLE  8. — CORRECTION  FACTORS  FOR  TIME  CALF  is  CARRIED 

To  Convert  to  305-Day  Farrow  Basis 
P  —  time  in  days  that  calf  is  carried,  commencing  at  conception. 


P 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

20... 

1.0015 

1.0015 

1.0016 

1.0017 

1.0017 

1.0018 

1.0019 

1.0019 

1.0020 

1.0021 

21  

1.0022 

1.0023 

1.0024 

1.0025 

1.0025 

1.0026 

1.0027 

1.0028 

1.0029 

1.0030 

22  

1.0031 

1.0033 

1.0034 

1.0035 

1.0036 

1.0037 

1  .  0039 

1  .  0040 

1.0042 

1  .  0044 

23  

1.0045 

1.0046 

1.0048 

1.0050 

1.0052 

1.0054 

1.0056 

1  .  0058 

1.0060 

1.0062 

24  

1.0065 

1.0067 

1.0069 

1.0072 

1.0075 

1.0077 

1.0080 

1.0083 

1.0086 

1.0089 

25  

1.0093 

1.0096 

1.0100 

1.0104 

1.0107 

1.0111 

1.0115 

1.0119 

1.0124 

1.0128 

26  

1.0133 

1.0138 

1.0144 

1.0149 

1.0155 

1.0160 

1.0166 

1.0173 

1.0179 

1.0185 

27  

1.0192 

1.0199 

1  .  0207 

1.0215 

1  .  0223 

1.0231 

1  .  0240 

1  .  0249 

1.0258 

1  .  0268 

28  

1  .  0278 

SUMMARY 

The  study  is  based  on  the  calendar  month  records  of  4,522  yearly 
records  of  the  Guernsey  Advanced  Registry.  The  records  have  been 
handled  statistically  to  secure  the  monthly  yields  for  farrow  cows  and 
cows  bred  at  monthly  intervals  following  parturition.  Yield  has  been 
measured  in  terms  of  4-percent  milk  (F.C.M.)  on  a  gross  energy  basis 
(one  pound  F.C.M.  =  331  large  calories),  estimated  as  AM  +  15F, 
(M  =  milk,  F  =  fat,  all  in  pounds). 

Equations  have  been  derived  expressing  the  rate  of  milk  secretion 

dy 

of  the  general  type  --  =  ae~kt  for  farrow  cows  and  cows  carrying  the 
at 

Sift  , 

calf  five  months  or  less;  and  —  =  ae~kt  —  beK(t  '  for  cows  carry- 
ing the  calf  more  than  five  months  (y  =  yield,  t  =  time  in  months  from 
calving,  and  c  =  time  in  months  from  calving  to  conception) .  In  these 
equations  a  is  representative  of  the  initial  level  of  production  and  k  is 
representative  of  persistency  of  milk  flow,  being  an  inverse  measure 
of  this  characteristic.  The  minus  term  of  the  second  equation  measures 
the  effect  of  pregnancy  on  milk  yield. 

With  305  days  as  the  standard,  the  ratio  of  the  305-day  record  to 
that  of  records  of  other  lengths  was  derived  from  the  corresponding 
definite  integrals  of  the  first  equation.  On  the  basis  of  the  average 


1926\      MILK  SECRETION  WITH  ADVANCE  IN  LACTATION  AND  GESTATION  35 

value  of  k,  .04412,  and  starting  at  the  same  time  after  calving,  the 
ratios  for  records  varying  from  200  to  365  days  in  length  have  been 
computed  and  tabulated  by  intervals  of  one  day.  The  305-day  record 
is  thus  equal  to  86.97  percent  of  the  365-day  record.  The  value  of  k 
is  associated  to  some  extent  with  age  and  productive  level.  Younger 
cows  have  a  lower  value  of  k  than  older  cows,  that  is,  the  younger  cows 
are  more  "persistent"  milkers. 

The  influence  of  pregnancy  on  yield  is  regarded  as  caused  directly 
by  a  physiological  inhibitor  to  milk  secretion  in  the  circulation,  rather 
than  as  due  indirectly  to  the  draft  of  the  growing  fetus  on  the  nutrients 
of  the  blood. 

The  value  found  for  K  was  very  consistently  1.09861.  The  value  of 
6  was  roughly  proportional  to  a  (b  =  .0000 la).  On  this  basis  the  ratio 
of  the  305-day  record  of  farrow  cows  to  the  305-day  record  of  gestating 
cows  has  been  computed  and  tabulated  for  lengths  of  pregnancy  varying 
from  200  to  280  days,  by  intervals  of  one  day.  The  average  decrease  in 
yield  for  the  first  5  months  of  pregnancy,  by  the  equation,  is  2.5  pounds, 
and  for  the  gestation  period  (9.2  months)  256  pounds  F.C.M.,  or 
85  therms. 


36  BULLETIN  No.  272 

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